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Identity and Search in Social Networks D.J.Watts, P.S. Dodds, M.E.J. Newman

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Maryam Fazel-Zarandi. Identity and Search in Social Networks D.J.Watts, P.S. Dodds, M.E.J. Newman. Outlines. Introduction The Hierarchical Model Discussion. Introduction. Source. Milgram’s Experiment. Short chains of acquaintances exist.

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Presentation Transcript
outlines
Outlines
  • Introduction
  • The Hierarchical Model
  • Discussion
milgram s experiment

Source

Milgram’s Experiment
  • Short chains of acquaintances exist.
  • People are able to findthese chains using only local information.
results in literature
Results in Literature
  • Connected random networks have short average path lengths:

xij log(N)

    • N = population size,
    • xij = distance between nodes i and j.
results in literature1
Results in Literature
  • Kleinberg (2000) demonstrated that emergence of the second phenomenon requires special topological structure.
  • For each node i:
    • local edges d(i,j) ≤ p
    • long-range directed edges

to q random nodes

Pr(ij) ~ d(i,j)-a

results in literature2
Results in Literature
  • If networks have a certain fraction of hubs can also search well.
  • Basic idea: get to hubs first
  • Hubs in social networks are limited.
hierarchical model why how
Hierarchical Model – Why? How?
  • Basic idea: impose some high-level structure, and fill in details at random.
  • Incorporate identity.
  • Need some measure of distance between individuals.
  • Some possible knowledge:
    • Target\'s identity, friends\' identities, friends\' popularity, where the message has been.
hierarchical network construction
Hierarchical Network Construction
  • xij = the height of the lowest common ancestor

level between i and j

  • z connections for each node with probability:

p(x) = ce-αx

Network constructed from template

Hierarchical template for the network

hierarchical network construction1
Hierarchical Network Construction
  • Individuals hierarchically partition the social world in more than one way.
    • h = 1, …, H hierarchies
  • Identity vector
    • is position of node i in hierarchy h.
  • Social distance:
directing messages
Directing Messages
  • At each step the holder i of the message passes it to one of its friends who is closest to the target t in terms of social distance.
  • Individuals know the identity vectors of:
    • themselves,
    • their friends,
    • the target.
expected number of steps
Expected Number of Steps
  • What is the expected number of steps to forward a message from a random source to a random target?
  • Define q as probability of an arbitrary message chain reaching a target.
  • Searchable network: Any network for which

q≥ r

for a desired r.

number of steps results
Number of Steps - Results
  • If message chains fail at each node with probability p, require

where L = length of message chain.

  • Approximation:

L ln r / ln (1 - p)

q = (1 - p)L ≥ r

searchable network regions
Searchable Network Regions
  • In H-αspace
  • p = 0.25, r = 0.05
  • b = 2
  • g = 100, z = 99
  • N=102400
  • N=204800
  • N=409600
probability of message completion
Probability of Message Completion
  • α = 0 (squares) versus α= 2 (circles)
  • N = 102400
  • q ≥ r

q < r

r = 0.05

milgram s data
Milgram\'s Data
  • N = 108
  • b = 10
  • g = 100
  • z = 300
  • Lmodel 6.7
  • Ldata 6.5
  • α = 1, H = 2
is this an acceptable model
Is this an acceptable model?
  • Simple greedy algorithm.
  • Represents properties present in real social networks:
    • Considers local clustering.
    • Reflects the notion of locality.
  • High-level structure + random links.
can the model be extended
Can the Model be Extended?
  • Should we consider other parameters such as friend’s popularity information in addition to homophily?
    • Allow variation in node degrees?
  • Assume correlation between hierarchies?
  • Are all hierarchies equally important?
applications
Applications
  • Can solutions to sociology problems inform other areas of research?
  • Suggested applications:
    • Construction of peer-to-peer networks.
    • Search in databases.
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